\newproblem{lay:6_1_30}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.1.30}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $W$ be a subspace of $\mathbb{R}^n$, and let $W^\perp$ be the set of all vectors ortohogonal to $W$. Show that $W^\perp$ is a subspace of $\mathbb{R}^n$ using
	the following steps:
	\begin{enumerate}[a.]
		\item Take $\mathbf{z}\in W^\perp$ and let $\mathbf{u}$ represent any vector in $W$. Then, $\mathbf{z}\cdot\mathbf{u}=0$. Take any scalar $c$ and show that $c\mathbf{z}$
				  is orthogonal to $\mathbf{u}$. (Since $\mathbf{u}$ is any arbitrary vector in $W$, this will show that $c\mathbf{z}$ is in $W^\perp$.)
		\item Take $\mathbf{z}_1,\mathbf{z}_2\in W^\perp$, and let $\mathbf{u}$ be any vector in $W$. Show that $\mathbf{z}_1+\mathbf{z}_2$ is orthogonal to $\mathbf{u}$. What
		      can you conclude about $\mathbf{z}_1+\mathbf{z}_2$? Why?
		\item Finish the proof that $W^\perp$ is a subspace of $\mathbb{R}^n$
	\end{enumerate}
}{
   % Solution
	\begin{enumerate}[a.]
		\item Let us calculate $(c\mathbf{z})\cdot\mathbf{u}$\\
		      \begin{center}
						$(c\mathbf{z})\cdot\mathbf{u}=c(\mathbf{z}\cdot\mathbf{u})=c\cdot 0=0$
					\end{center}
					So $c\mathbf{z}$ is orthogonal to any vector $\mathbf{u}$ in $W$, and consequently $c\mathbf{z}$ belongs to $W^\perp$.
		\item Let us calculate $(\mathbf{z}_1+\mathbf{z}_2)\cdot\mathbf{u}$\\
		      \begin{center}
						$(\mathbf{z}_1+\mathbf{z}_2)\cdot\mathbf{u}=\mathbf{z}_1\cdot\mathbf{u}+\mathbf{z}_2\cdot\mathbf{u}=0+0=0$
					\end{center}
					So $\mathbf{z}_1+\mathbf{z}_2$ is orthogonal to any vector $\mathbf{u}$ in $W$, and consequently $\mathbf{z}_1+\mathbf{z}_2$ belongs to $W^\perp$.
		\item We still need to show that $\mathbf{0}\in W^\perp$ \\
		      \begin{center}
						$\mathbf{0}\cdot\mathbf{u}=0$
					\end{center}
					So $\mathbf{0}\in W^\perp$.
	\end{enumerate}
}
\useproblem{lay:6_1_30}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
